[Math] Find matrix $T$ relative to standard basis

Question

problem:

A linear transformation $T$ rotates each vector in $\Bbb R^2 $ clockwise through $90$ degree

Find matrix $T$ relative to standard basis$
\begin{bmatrix}
1 \\
0 \\
\end{bmatrix}
$,
$
\begin{bmatrix}
0 \\
1 \\
\end{bmatrix}
$

Solution: Here value of linear transformation is not given

like $T(a,b)=(a,0)$

I don't know "how to solve such question"

Answer 1

Since we have

$$T\begin{bmatrix} 1 \\ 0 \\ \end{bmatrix}=\begin{bmatrix} 0 \\ -1 \\ \end{bmatrix}\quad\text{and}\quad T\begin{bmatrix} 0 \\ 1 \\ \end{bmatrix}=\begin{bmatrix} 1 \\ 0 \\ \end{bmatrix}$$ so the matrix of $T$ in the standard basis is $$\begin{bmatrix} 0&1 \\ -1&0 \\ \end{bmatrix}$$